Sample Questions
Question 1
Easy
An actuary records five independent claim severities (in thousands):
Calculate the unbiased sample variance.
Calculate the unbiased sample variance.
Solution
E is correct. First compute the sample mean: The deviations from the mean are , with squared deviations , summing to . The unbiased sample variance uses the divisor : The value 29 falls in the interval , so the answer is E.
Question 2
Medium
Let be a random sample of five independent observations from a continuous Uniform distribution. Calculate the probability that the sample maximum exceeds 8.
Solution
B is correct. The maximum is at most 8 if and only if every observation is at most 8. By independence, Therefore . This value falls in the interval .
Question 3
Hard
An actuary observes six iid claim amounts drawn from an exponential distribution with mean . Let denote the order statistics.
Calculate
Calculate
Solution
B is correct.
Step 1: Translate the event in terms of how many observations fall on each side of 200. means the third-smallest of the six exceeds 200, equivalently at most two of the six observations are
Step 2: Define a binomial counter. Let . Since the are iid, with
Step 3: Compute using the exponential cdf.
Step 4: Sum the binomial probabilities for .
Step 5: Add.
The single-observation tail is a tempting shortcut but ignores the order-statistic structure entirely; it answers a different question.
Step 1: Translate the event in terms of how many observations fall on each side of 200. means the third-smallest of the six exceeds 200, equivalently at most two of the six observations are
Step 2: Define a binomial counter. Let . Since the are iid, with
Step 3: Compute using the exponential cdf.
Step 4: Sum the binomial probabilities for .
Step 5: Add.
The single-observation tail is a tempting shortcut but ignores the order-statistic structure entirely; it answers a different question.
